Intervals in the Hales-Jewett theorem
- The Hales–Jewett theorem states that for any m and r there exists an n such that any r-colouring of the elements of [m]n contains a monochromatic combinatorial line. We study the structure of the wildcard set S ⊆ [n] which determines this monochromatic line, showing that when r is odd there are r-colourings of n where the wildcard set of a monochromatic line cannot be the union of fewer than r intervals. This is tight, as for n sufficiently large there are always monochromatic lines whose wildcard set is the union of at most r intervals.
- Publication status:
- Peer review status:
- Peer reviewed
- This paper has been accepted for publication in the European Journal of Combinatorics.